Explore. Three decimal places
Two copies of the boy’s evening at the lake run side by side on your own device, in perfect lockstep and with the same arithmetic. Make one difference to the copy, too small for anyone in that world ever to measure, and watch where it leads.
Precision, the subtlest limit, leaves the machine able to carry a world while changing which world gets carried.
Two evenings
- The wind this evening
- In its copy
Consider two copies of a world run in lockstep, every state of one identical to the corresponding state of the other.
Chapter 2, Limits and Seams, ¶ 40
Lockstep copies require arithmetic that is reproducible bit for bit, which is achievable and has to be chosen deliberately.
The copy was retyped from a printout that kept three decimal places, as Lorenz’s did.
The printout had rounded each number to three decimal places, while the machine had carried six. When Lorenz came back, the new forecast had drifted from the old one and then departed from it entirely, as though the two runs described different planets.
The copy’s laws were written with one constant changed in its sixth decimal place, 10.000001 where the evening has 10.
A designer who writes down the strength of a force has written a particular number to a particular number of digits, and in a chaotic world the digits beyond anything the inhabitants will ever measure still decide which histories occur.
The copy adds the same numbers at every step, in another order, as a machine split across many processors might.
On real hardware, two copies of the boy’s evening started from the same state could diverge over nothing more than the order of a sum, and chaos would then carry them apart into different histories.
A difference in the sixth decimal place of one quantity, introduced by the substrate’s arithmetic, grows until it is a different evening. The stone skips five times instead of four, the boy laughs a moment later, and the remainder of his life unfolds from a slightly different start.
The leak
A designer may write a physics in which energy is conserved exactly, and a machine carrying that physics by a careless method of stepping forward in time will let energy drift a little with every step.
Chapter 2, Limits and Seams, ¶ 39
Four floats rise and fall on still water under a law that keeps their energy exactly the same forever. Each float carries that law forward in time by a different method.
-
As the law is written
The motion the law describes, worked out exactly.
Energy now as at the start
-
A careless method
Each step uses only where the float was, and overshoots a little.
Energy now as at the start
-
Careless the other way
Each step uses only where the float will be, and falls short a little.
Energy now as at the start
-
A symplectic method
Each step updates speed, then position from the new speed, keeping the error bounded.
Energy now as at the start
The rings on the lake might never quite die away, or might die faster than the laws say they should, and the world’s physicists, if they measured carefully enough, would discover that their conservation law is false.
Chapter 2, Limits and Seams, ¶ 39
Symplectic integrators preserve the geometric structure of the laws so that energy errors stay bounded over very long spans, and schemes built on exact integer arithmetic can be made perfectly reversible, conserving what the laws conserve to the last bit.
The gap
The lesson these cases share is that the inhabitants live under the physics that is actually computed, and the designer answers for any gap between it and the physics that was written.
Chapter 2, Limits and Seams, ¶ 41
One more digit in the machine’s arithmetic might have held the stone to four.
How the evenings are computed
Both evenings are computed in your browser with the same 64-bit arithmetic. The weather of the evening follows the three equations Lorenz published in 1963, with his values for their constants (10, 28, and 8/3), stepped forward by the fourth-order Runge–Kutta method in steps of a two-hundredth of a minute. Every four minutes the boy throws a stone, and the first of the three numbers, read as the wind across the lake, sets how many times it skips, from three to six, a following wind carrying it farther. The wind and the skipping are this page’s invention, and the arithmetic is real. Until you make a difference, the two copies agree to the last binary digit, because they perform the same operations in the same order. The floats follow a law of simple oscillation that keeps their energy constant, and each is stepped forward by the method named beside it.